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A '''residue number system''' ('''RNS''') represents a large integer using a set of smaller integers, so that [[computation]] may be performed more efficiently. It relies on the [[Chinese remainder theorem]] of [[modular arithmetic]] for its operation, a mathematical idea from [[Sun Tzu (mathematician)|Sun Tsu Suan-Ching]] (Master Sun’s Arithmetic Manual) in the 4th century AD.
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==Defining a residue number system==
 
A residue number system is defined by a set of ''N'' integer constants,
 
:{''m''<sub>1</sub>, ''m''<sub>2</sub>, ''m''<sub>3</sub>, ... , ''m''<sub>''N''</sub> },
 
referred to as the ''moduli''. Let ''M'' be the [[least common multiple]] of all the ''m''<sub>''i''</sub>.
 
Any arbitrary integer ''X'' smaller than ''M'' can be represented in the defined residue number system as a set of ''N'' smaller integers
 
:{''x''<sub>1</sub>, ''x''<sub>2</sub>, ''x''<sub>3</sub>, ... , ''x''<sub>''N''</sub>}
 
with
 
:''x''<sub>''i''</sub> = ''X'' ''[[Modulo operation|modulo]]'' ''m''<sub>''i''</sub>
 
representing the residue class of ''X'' to that modulus.
 
Note that for maximum representational efficiency it is imperative that all the moduli are [[coprime]]; that is, no modulus may have a common factor with any other. ''M'' is then the product of all the ''m''<sub>i</sub>.
 
For example RNS(4|2) has non-coprime moduli, resulting in the same representation for different values.<ref>Parhami, Computer Arithmetic, Algorithms and Hardware Design</ref>
 
  (3)decimal = (3|1)<sub>RNS(4|2)</sub>
  (7)decimal = (3|1)<sub>RNS(4|2)</sub>
 
==Operations on RNS numbers==
 
Once represented in RNS, many arithmetic operations can be efficiently performed on the encoded integer.  For the following operations, consider two integers, ''A'' and ''B'', represented by ''a''<sub>''i''</sub> and ''b''<sub>''i''</sub> in an RNS system defined by ''m''<sub>''i''</sub> (for ''i'' from 0 ≤ ''i'' ≤ ''N'').
 
===Addition and subtraction===
 
Addition (or subtraction) can be accomplished by simply adding (or subtracting) the small integer values, modulo their specific moduli. That is,
:<math>C=A\pm B \mod M</math>
can be calculated in RNS as
:<math>c_i=a_i\pm b_i \mod m_i</math>
 
One has to check for [[arithmetic overflow|overflow]] in these operations.
 
===Multiplication===
 
Multiplication can be accomplished in a manner similar to addition and subtraction.  To calculate
:<math>C = A \cdot B  \mod M, </math>
we can calculate:
:<math>c_i = a_i\cdot b_i \mod m_i</math>
Again overflows are possible.
 
===Division===
 
Division in residue number systems is problematic.  A paper describing one possible algorithm is available at [http://www.cs.rpi.edu/research/ps/93-9.ps]. On the other hand, if ''B'' is coprime with ''M'' (that is <math>b_i\not =0</math>) then
:<math>C=A\cdot B^{-1} \mod M</math>
can be easily calculated by
:<math>c_i=a_i \cdot b_i^{-1} \mod m_i</math>
where <math>B^{-1}</math> is [[multiplicative inverse]] of ''B'' modulo ''M'', and <math>b_i^{-1}</math> is multiplicative inverse of <math>b_i</math> modulo <math>m_i</math>.
 
==Practical applications==
 
RNS have applications in the field of [[Digital data|digital]] [[computer arithmetic]]. By decomposing in this a large integer into a set of smaller integers, a large calculation can be performed as a series of smaller calculations that can be performed independently and in parallel.
 
==Integer factorization==
The RNS can improve efficiency of [[trial division]]. Let <math>X=Y\cdot Z</math> a [[semiprime]]. Let <math>m_1=2, m_2=3, m_3=5,\dots</math> represent first ''N'' primes. Assume that <math>Y>m_N</math>, <math>Z>m_N</math>. Then <math>x_i=y_i\cdot z_i</math>, where <math>x_i\not = 0</math>. The method of trial division is the method of exhaustion, and the RNS automatically eliminates all ''Y'' and ''Z'' such that <math>y_i=0</math> or <math>z_i=0</math>, that is we only need to check
:<math>\prod_{i=1}^N(m_i-1)=M\prod_{i=1}^N\left(1-\frac{1}{m_i}\right)</math>
numbers below ''M''. For example, ''N'' = 3, the RNS can automatically eliminate all numbers but
 
:1,7,11,13,17,19,23,29 mod 30
 
or 73% of numbers. For ''N'' = 25 when <math>m_i</math> are all prime numbers below 100, the RNS eliminates about 88% of numbers. One can see from the above formula the diminishing returns from the larger sets of moduli.
 
==Associated mixed radix system==
A number given by <math>\{x_1,x_2,x_3,\ldots,x_n\}</math> in the RNS can be naturally represented in the ''associated [[mixed radix]] system'' (AMRS)
 
:<math>X=\sum_{i=1}^Nx_iM_{i-1}=x_1+m_1(x_2+m_2(\cdots+m_{N-1}x_{N})\cdots),</math>
where
:<math>M_0=1,M_i=\prod_{j=1}^i m_j</math> for <math>i>0</math> and <math>0\leq x_i<m_i.</math>
 
Note that after conversion from the RNS to AMRS, the comparison of numbers becomes straightforward.
 
==See also==
* [[Covering system]]
* [[Reduced residue system]]
 
==References==
 
<references/>
* [http://hal-lirmm.ccsd.cnrs.fr/docs/00/10/64/70/PDF/D547.PDF Efficient RNS bases for Cryptography]  // IMACS'05: World Congress: Scientific ComputationApplied Mathematics and Simulation. 2005.
[[Category:Modular arithmetic]]
[[Category:Computer arithmetic]]

Latest revision as of 14:55, 8 November 2014

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